TITLE: Lindelöf and second countable spaces
QUESTION [7 upvotes]: Can anyone give me some examples and non examples of Lindelöf or second countable space and spaces that is Lindelöf but not second countable?   And I understand the definition but find it is hard to visualize and imagine.
I have tried google it but it turns out I only found some silly examples like finite set or empty set.
In general, how can one construct a topological space that is Lindelöf or second countable?
Someone in stack exchange said the real line with discrete topology is Lindelöf, but I do not think so. We can simply construct an open cover defined by the collection of all the singleton set. And this open cover is well defined since singleton set is open in discrete topology. Hence, by definition it is not Lindelöf.
Last question, is (0,1) in the real line equipped with usual topology Lindelöf? I think it is Lindelöf but I could not give any formal proof.
(0,1) fails to be a compact set since we can construct an open cover defined by (1/n,1-1/n) but this open cover does not work so well for arguing for Lindelöf property since quotient number is dense in (0,1). So intuitively I think it is Lindelöf.
I wrote a pretty long question. My mothertongue is not English. Hopefully, you guys can understand me.

REPLY [2 votes]: Lindelöf and second countable are saying that a space is "small" in some sense; so one way to find non-examples is to take products of lots of spaces, such products (or powers) are "big".
$\Bbb R^I$ is not Lindelöf for $I$ uncountable. (it's also not first countable at any point). It is not normal (which is one of the easier ways to see it's not Lindelöf). Of course you're right that the discrete reals are not Lindelöf (take the open cover by singleton sets). BTW, it's a non-trivial fact that if $I$ has size at most that of $\Bbb R$, this product is still separable, so it's also an example of a separable non-Lindelöf space for such $I$.
$[0,1]^I$ for $I$ uncountable is compact (Tychonoff's theorem) so Lindelöf but not first countable at any point too, so certainly not second countable either. But it is normal, of course. And separable iff $|I| \le |\Bbb R|$.
For metric spaces: Lindelöf, second countable and separable are equivalent properties (see a more general fact in my answer here. So $(0,1)$ in the usual topology is certainly Lindelöf, as the rationals in it are dense.